1

On the Robust Control of the Boost Converter H. Sira-Ram´ırez †, M. A. Oliver-Salazar‡, J.A. V´azquez-Santacruz†, M. Velasco-Villa† †Centro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico Nacional. Departamento de Ingenier´ıa El´ectrica, Secci´on de Mecatr´onica. Apartado postal 14740, M´exico DF C.P. 07360 ‡Centro Nacional de Investigaci´on y Desarrollo Tecnol´ogico Coordinaci´on de Mecatr´onica Cuernavaca, Morelos, M´exico

Abstract—In this article, we propose a linear Total Active Disturbance Rejection approach for the flatness based control of an uncertain “Boost” DC-to-DC power converter subject to unknown load current demands of arbitrary time-varying nature. The approach makes use of two Generalized Proportional Integral (GPI) observers; one for the precise determination of the load current, and a second one for the on-line determination of the additive nonlinearities affecting the stored energy second order dynamics. After disturbance suppression, a certainty equivalence controller completes the controller design by disturbancedependent control input gain cancelation. Simulations demonstrate the effectiveness of the control approach in the tracking of a desired stored energy profile.

I. I NTRODUCTION Controllers based on disturbance estimation, followed by its direct cancelation from the control actions, have been developed along different, but closely related, directions. In this respect, the outstanding work of professor C.D. Johnson known as Disturbance Accommodation Control (DAC), dates from the nineteen seventies (see [14]). The evolution of the DAC approach is surveyed in [15]. Related developments, known as Active Disturbance Rejection (ADR), are addressed in the work of the late professor Jingqing Han [10]. This includes nonlinear observers, with side developments related to: efficient time derivatives computation, practical relative degree calculation and nonlinear PID control extensions. Professor Z. Gao, and his colleagues (see [8], [9], [22], [23]) have made important contributions in ADR and have proposed a new control paradigm, closely related to the vision of Prof. Han. Prof. M. Fliess and C. Join (see [4] and [7]), proposed Intelligent PID Control (IPIDC) related to Prof. Han’s handling of state dependent perturbation inputs. The method is based on Differential Algebra and it implies to resort to low order, phenomenological plant models. The effect of nonlinear uncertain additive terms, locally modeled as timepolynomials that need to be updated, is suitably annihilated. Time-varying state dependent control gains are deemed as piece-wise constant gains which can be efficiently on line identified (see [5] for a sound theoretical basis). Recently, an entirely robust linear approach has been proposed for the control of uncertain non-linear differentially flat systems, using an input-output approach (see [18]) which may be considered as the analog counterpart of the algebraic methods stated in [5]. Linear GPI observers are used to locally estimate, in

a self-updating manner, the joint effects of exogenous and endogenous additive disturbances, while the control input gain is assumed to be known and, at most, output dependent ([19]). The approach, however, has significant implications in nonlinear chaotic systems estimation as demonstrated in [16]. The GPI observer based approach has also been extended to the control of nonlinear multi-variable systems exhibiting known time delays in the control inputs (see [21]). In several boost converter applications, load current variations need special attention. In [12], an adaptive controller is designed to provide robustness against perturbations of the output voltage. In [11] a voltage compensator is developed based on extended linearization and gain scheduling with a nonlinear current mode control. A control strategy is presented in [1] based on the combination of sliding mode control and the current mode control for line and load regulation with low current ripple. In [2] a parallel damping (instead of series damping) passivity-based control stategy is presented to achieve robustness against load variations in the buck and boost converters. The study presented in [13] is an input-output feedback linearization based scheme, taking the inductor current as the controlled output. In this article, we propose a GPI observer-based ADR approach for controlling a perturbed “Boost” DC-to-DC power converter in a total stored energy trajectory tracking task with a time-varying, unknown, load demand. We design a robust GPI observer-based controller, with certainty equivalence input gain cancelation, using the average converter model. The average design is translated into a switched controlled design using the Sigma-Delta modulation implementation alternative. Under the assumption of full state measurement, a GPI observer estimates, in a rather precise fashion, the unknown load current. This estimation is carried out on the basis of the output voltage dynamics. The nonlinear additive terms affecting the total stored energy second order dynamics are also linearly estimated using a GPI observer. The load current dependent control input gain is canceled within a certainty equivalent PID controller scheme. Section II deals with a general formulation of the ADR control problem in nonlinear differentially flat systems. In section III, we present the GPI observer based ADR for the total stored energy trajectory tracking task in a Boost converter subject to unknown load current demands. Section IV presents simulation results assessing the effectiveness of the proposed

2

control scheme. The conclusions and suggestions for further research are collected at the end of the article. II. ACTIVE D ISTURBANCE R EJECTION IN N ONLINEAR D IFFERENTIALLY F LAT S YSTEMS Consider the single-input, single-output, smooth nonlinear n-dimensional input-output perturbed system, y (n)

=

˙ ψ(t, y, y, ˙ ..., y (n−1) , ζ(t), ζ(t)) +φ(t, y, y, ˙ · · · , y (n−1) , ζ(t))u

(1)

The unperturbed system is, therefore, differentially flat, with flat output, y. We say that the system is perturbed differentially flat, whenever ζ(t) is not identically zero. In this last case, the state components in the vector, x, and the control input, u, are differentially parameterizable in terms of, both, the output, y, and the exogenous perturbation input ζ(t). Conversely, the phase variables (y, y, ˙ ..., y (n−1) ) may be written in terms of the n-dimensional state of the system, denoted by x, the perturbation input, ζ, and a finite number of its time derivatives. The above input-output description of the system typically corresponds to non-matched disturbances in the state space formulation1 . (n−1) ˙ The term: ψ(t, y(t), y(t),...,y ˙ (t), ζ(t), ζ(t)), is assumed to be uniformly absolutely bounded for any suitable control input, u, achieving the control objectives. The nonlinear gain function, φ(t, y(t), y(t), ˙ · · · , y (n−1) (t), ζ(t)), is known and also assumed to be bounded and uniformly bounded away from zero, i.e., there exists a strictly positive constants, µ, and, β, such that: supt |φ(t, y(t), y(t), ˙ · · · , y (n−1) (t), ζ(t))| ≤ β and µ ≤ inf t |φ(t, y(t), y(t), ˙ · · · , y (n−1) (t), ζ(t))|, for all smooth, bounded solutions, y(t), of (1) obtained with a suitable control input u achieving the control objectives. We formulate the problem as follows: Given a desired flat output reference trajectory, y ∗ (t), devise a linear output feedback controller for system (1) so that, regardless of the effect of the additive perturbation signal, ψ(·), the flat output, y, tracks the desired reference signal, y ∗ (t), even if in an approximate fashion. This approximate character specifically means that the tracking error, e(t) = y−y ∗ (t), and its first n-time derivatives, converge towards a small as desired neighborhood of the origin in the reference trajectory tracking error phase space in a globally asymptotically exponentially dominated manner. We denote by, ξ(t), the following expression, ξ(t) = ψ(t, y(t), y(t), ˙ ..., y (n−1) (t), ζ(t)) This quantity, deemed as a perturbation input, needs to be online estimated for proper subsequent cancelation at the controller stage. The signal, ξ(t), viewed from the observer design perspective, represents an exogenous time-varying quantity which is easily shown to be observable in the sense of 1 Specifically the state representation is of the form: x ˙ = f (x) + g(x)u + h(x) 6= γ(x)ζ, y = h(x) with Lg Ljf h(x) = 0, j = 1, 2, ..., n−2, Lg Ln−1 f 0 on a sufficiently large open region of Rn containing an operating point, x0 . Further, Lγ Lkf h(x) = 0, k = 0, 1, ..., n − 3, Lγ Lif h(x) 6= 0, i = n − 2, n − 1. This is, precisely, the case of the Boost converter here treated.

Diop and Fliess [3]. Regarding this state dependent signal as an, unstructured, time-varying signal, is at the heart of the observer-based ADR approach. Our linear observer design strategy consists in approximately estimating this time-varying quantity, ξ(t), using an, instantaneous, internal time polynomial model, realized in the form of a chain of integrators of length, p − 1, at the observer stage for a fixed, sufficiently large integer p. The observer will also approximately estimate, in a simultaneous fashion, the phase variables associated to the system flat output y. When forcing the dominantly linear, perturbed, output estimation error dynamics to exhibit an asymptotically convergent behavior, the internal model for ξ(t) is automatically and continuously self-updated. We assume that ξ(t) and a finite number of its time derivatives, ξ (k) (t), are uniformly absolutely bounded, for k = 1, 2, ..., p, for some sufficiently large integer p. We have the following general result: Theorem 1: The GPI observer-based dynamical feedback controller: h u=

[y ∗ (t)](n) −

Pn−1 ¡ j=0

i ¢ ˆ κj [yj − (y ∗ (t))(j) ] − ξ(t)

φ(t, y, y, ˙ · · · , y (n−1) , ζ(t))

ˆ = z1 ξ(t) y˙ 1 = y2 + λp+n−1 (y − y1 ) y˙ 2 = y3 + λp+n−2 (y − y1 ) .. . y˙ n = φ(t, y)u + z1 + λp (y − y1 ) z˙1 = z2 + λp−1 (y − y1 ) .. . z˙p−1 = zp + λ1 (y − y1 ) z˙p = λ0 (y − y1 )

(2)

asymptotically exponentially drives the tracking error phase (k) variables, ey = y (k) − [y ∗ (t)](k) , k = 0, 1, .., n − 1 to an arbitrary small neighborhood of the origin, of the tracking error phase space, which can be made as small as desired from the appropriate choice of the controller gain parameters, {κ0 , ..., κn−1 }. Moreover, the estimation errors: e˜(i) = y (i) − yi+1 , i = 0, ..., n − 1 and the perturbation estimation errors: zj − ξ (j−1) (t), j = 1, ..., p, asymptotically exponentially converge towards a small as desired neighborhood of the origin of the reconstruction error space, with the appropriate choice of the observer gain parameters, {λ0 , ..., λp+n−1 }. Proof The proof is based on the fact that the estimation error e˜ satisfies the perturbed linear differential equation e˜(p+n) + λp+n−1 e˜(p+n−1) + · · · + λ0 e˜ = ξ (p) (t)

(3)

Since ξ (p) (t) is assumed to be uniformly absolutely bounded then there exists coefficients λk such that e˜ converges to a small vicinity of zero, provided the roots of the associated characteristic polynomial in the complex variable s: sp+n + λp+n−1 sp+n−1 + · · · + λ1 s + λ0

(4)

3

are all located deep into the left half of the complex plane. The further away from the imaginary axis, of the complex plane, are these roots located, the smaller the neighborhood of the origin, in the estimation error phase space, where the estimation error e˜ will remain uniformly ultimately bounded. Let l(˜ e, e˜˙ , ...˜ e(n−1) ) be a linear function of the estimation error, e˜, and its time derivatives, given by l(˜ e, e˜˙ , ...˜ e(n−1) ) =

n−1 X

kj e˜(j)

(5)

L

iL

E

Fig. 1.

u=1

vC

u=0

I(t)

C

R

Perturbed Boost Converter Circuit

j=0

The tracking error, ey = y − y ∗ (t), evolves according to the following linear perturbed dynamics (n−1) e(n) + · · · + κ0 ey y + κn−1 ey ˆ − l(t, e˜, e˜˙ , ...˜ = ξ(t) − ξ(t) e(n−1) )

Choosing the controller coefficients {κ0 , · · · , κn−1 }, so that the associated closed loop characteristic polynomial sn + κn−1 sn−1 + · · · + κ0

(6)

exhibits its roots sufficiently far from the imaginary axis in the left half portion of the complex plane, the tracking error, and its various time derivatives, are guaranteed to dominantly asymptotically exponentially converge towards a vicinity of the origin of the tracking error phase space. We also have that the disturbance estimation error satisfies: ξ(t) − ξˆ = ξ(t) − z1 = r(t) = e˜(n) + λp e˜ Since e˜(n) and e˜ are forced to converge to a small as desired vicinity of the origin, the local residual term, r(t), of the polynomial approximation is uniformly absolutely bounded by a small neighborhood of zero. This fact clearly depicts the selfupdating character of the local time polynomial model of the lumped disturbance ξ(t).

where, x1 , is the normalized inductor current. p The variable, y = x2 , is the normalized output voltage, Q = R C/L, is the fixed normalized resistive output load, assumed to be perfectly known, and I(t) denotes the normalized time-varying load current. When, with an abuse of notation, the normalized equations (8) are written, substituting the switched input u by the continuous input, uav ∈ (0, 1), we refer to the controlled system as the average system model. The corresponding state variables x1 , x2 and the output y are deemed as the average state variables. We specifically state the following set of assumptions on the average system, 1) I(τ ) is strictly positive and uniformly bounded. Its first ˙ ), is continuous and uniformly order time derivative, I(τ absolutely bounded. 2) The normalized resistive load Q is constant and perfectly known. All other system parameters allowing the above normalization are perfectly known 3) The average input current, x1 , and the average output voltage y = x2 are measurable. We deal with the total normalized stored energy of the system, denoted by E, defined as

III. ROBUST C ONTROL OF THE T OTAL S TORED E NERGY IN A B OOST C ONVERTER

1 2 (x + x22 ) (9) 2 1 The total power, i.e., the first order time derivative of, E, is readily found to be given by

Consider the Boost converter system, shown in Figure 1, subject to an unknown, time-varying, external load current demand, I(t). The system equations are readily obtained as

x2 E˙ = x1 − 2 − x2 I(τ ) Q

diL dt dvC C dt η L

=

−uvC + E

=

uiL −

=

vC

vC − I(t) R

(7)

iL is the inductor current, vC is the output voltage, u ∈ {0, 1} is switched control input. η is the output voltage. All system parameters L, C, R and E are assumed to be perfectly known. We perform the following normalizing p state, and time, coordinate transformation: x = (i/E) L/C, x2 = v/E, 1 √ τ = t/ LC. The normalized equations, governing the converter, are given by (“ ˙ ” stands for d/dτ ): x˙ 1

=

x˙ 2

=

y

=

−ux2 + 1 x2 − I(τ ) ux1 − Q x2

(8)

E=

(10)

where the first term represents the input power (the product of the normalized input current and the unit normalized voltage source) the second term is the power dissipated in the resistive load and, the last term, represents the input power to the unknown non-regenerative load. The total stored energy variable, E, is relative degree 2 and, via some straightforward algebraic manipulation, is seen to satisfy the following average perturbed relation: ¸ · 2x1 x2 ¨ + x1 I(τ ) uav E = − x2 + Q 2x2 3x2 I(τ ) ˙ ) (11) +1 + 22 + + I 2 (t) − x2 I(τ Q Q The unperturbed Boost converter is differentially flat (see Fliess et al. [6] and, also, Sira-Ram´ırez and Agrawal [17]) with the total stored energy, E, as the flat output. The perturbed version of the system is differentially parameterizable in terms

4

of the flat output, the perturbation input and a finite number of its time derivatives. Clearly, the average total stored energy E satisfies a second order system of perturbed differential equations of the form: E˙1 E˙2

= =

E2 ˙ )) (12) α(E1 , E2 , I(τ ))uav + β(E1 , E2 , I(τ ), I(τ

with E1 = E and E2 = E˙ and, in terms of the states x1 , x2 , · ¸ 2x1 x2 ˙ α(E, E, I(τ )) = − x2 + + x1 I(τ ) Q 2 ˙ I(τ ), I(τ ˙ )) = 1 + 2x2 + 3x2 I(τ ) + I 2 (τ ) β(E, E, 2 Q Q ˙ −x2 I(τ ) The lack of knowledge of I(τ ), the need to compute E˙ and the difficulty in on-line synthesizing the nonlinearities α(·) and β(·) prompts us to formulate the problem as follows: A. Problem Formulation Given the average Boost converter circuit, (8), where all the previous assumptions are valid. It is desired to smoothly regulate the average total stored energy, E, from a given initial constant value: Einit , towards a final desired value, Ef inal , in a pre-specified amount of normalized time, T −τ0 , irrespectively of the, time-varying, magnitude of the load demand I(τ ) and based only on synthesis of the total stored energy E. B. A GPI Observer for the Load Current Consider the uncertain average output voltage equation with, y, representing the measured output voltage x2 . y y˙ = uav x1 − − I(τ ) (13) Q Definition 2: A signal, z(τ ), is said to be ultimately arbitrarily close to a signal, ζ(τ ), denoted by z(τ ) ≈ ζ(τ ), whenever given a small as desired constant, ² > 0, for which there exists a finite time, t² > 0, such that, the error signal, e(τ ) = z(τ ) − ζ(τ ), satisfies: supτ ≥τ² |e(τ )| ≤ ². In other words, the error e(τ ) is said to be uniformly absolutely ultimately bounded by an open spherical neighborhood of zero, of radius ². We have the following result: Lemma 3: Let m be a sufficiently large integer. The GPI observer, yˆ yˆ˙ = uav x1 − + z1 + γm (y − yˆ) Q z˙1 = z2 + γm−1 (y − yˆ) z˙2 = z3 + γm−2 (y − yˆ) .. . z˙m

= γ0 (y − yˆ)

(14)

produces a redundant estimate, yˆ, of the measured output variable, y, and an estimate, −z1 , of the unknown, timevarying, load current I(τ ) which are ultimately arbitrarily close to the their actual values, y(τ ), and, I(τ ), provided

the set of coefficients: {γ0 , . . . , γm }, of the characteristic polynomial pI (s), given by: 1 pI (s) = sm+1 + (γm + )sm + γm−1 sm−1 + · · · + γ1 s + γ0 Q (15) are chosen such that the roots of, pI (s), are located sufficiently far into the left half of the complex plane. Proof (m) Let ey = e1 = y − yˆ, e˙ y = e2 = y˙ − yˆ˙ ,..., ey = em+1 = (m) (m) y − yˆ denote the phase variables of the reconstruction error dynamics. From (13) and (14), the estimation, or reconstruction, error vector dynamics is then given by 1 e˙ 1 = −z1 − I(τ ) − (γm + )e1 Q z˙1 = z2 + γm−1 e1 .. . z˙m = γ0 e1 (16) Upon elimination, through straightforward differentiation, of the z variables in (16), the output voltage reconstruction error ey = y − yˆ is readily seen to satisfy the following linear, injected, perturbed, scalar differential equation: 1 e(m+1) + (γm + )e(m) + γm−1 e(m−1) + ··· y y Q y (17) · · · + γ1 e˙ y + γ0 ey = −I (m) (τ ) Since, by assumption I (m) (τ ) is L∞ then there exists a strictly positive constant K such that supt |I (m) (τ )| ≤ K. Define the (m) vector: χ = (ey , e˙ y , · · · , ey )T . The disturbed system has the (m) form: χ˙ = Aχ + bI (τ ), with A being a stable matrix in companion form, and b a vector with zero elements, except for its last component which has value equals to −1. The matrix, Q = A + AT , is symmetric and negative definite, with real negative eigenvalues, denoted as λ(Q). Let λmax (Q) be the eigenvalue with the minimum absolute value. The Lyapunov function candidate, V (x) = 21 χT χ = 12 kχk2 , satisfies: 1 T χ (A + AT )χ − χT I (m) (τ ) 2 1 ≤ − |λmax (Q)|kχk2 + Kkχk 2 The spherical region: V˙ (χ) =

S = {χ ∈ Rm+1 | kχk2 ≤ 4K 2 /|λmax (Q)|2 },

(18)

defines the convergence region of all the reconstruction error trajectories in the reconstruction error phase space, (ey , e˙y , · · · , ey (m) ). As the distance between the eigenvalues of the matrix A (i.e, the roots of p(s)) and the imaginary axis is increased, the radius of the sphere S is reduced, reducing the region of convergence of the injected reconstruction error trajectories. Consider now the equation satisfied by ey in terms of z1 . We have 1 e˙ 1 = −z1 − I(τ ) − (γm + )e1 (19) Q Since e1 = ey and e˙ 1 = e2 are ultimately absolutely bounded by a small as desired ball around the origin, then, from the

5

1 above relation, z1 = −I(τ ) − (γm + Q )e1 − e2 . Hence, the difference between z1 and −I(τ ) is also uniformly absolutely ultimately bounded by a small as desired sphere located at the origin. z1 is then ultimately arbitrarily close to −I(τ ). By the same token, from the differential equation for z2 , namely: ˙ )−γm−1 e1 . Since z˙1 = z2 +γm−1 e1 , it follows that z2 ≈ −I(τ e1 is uniformly ultimately bounded to a small as desired region of the origin of the estimation error phase space, it follows that ˙ ). In general zj (τ ) ≈ z2 is ultimately arbitrarily close to −I(τ (j−1) −I (τ ), j = 1, 2, ... in the above sense. Arbitrarily close estimation of the unknown load current −I(τ ) allows the use of its arbitrarily close estimate z1 in a certainty equivalence controller design for the total stored energy E.

C. A Linear GPI Observer Based Active Disturbance Rejection Controller for the Total Stored Energy Consider the average total stored energy dynamics, E˙1 E˙2

= =

E2 (20) ˙ )) α(E1 , E2 , I(τ ))uav + β(E1 , E2 , I(τ ), I(τ

Since E, and its time derivative, are known, thanks to state availability and the GPI observer based gathered knowledge of the perturbation input current demand, I(τ ); the control ˙ I(τ )), can be efficiently canceled from input gain, α(E, E, ˆ )). The additive the certainty equivalence gain: α(E1 , E2 , I(τ ˙ )), may be considered perturbation term, β(E1 , E2 , I(τ ), I(τ to be unknown, and being an exogenous signal to the GPI ˙ )), can observer, its time realization, β(E1 (τ ), E2 (τ ), I(τ ), I(τ be linearly estimated, in a lumped manner, using a GPI ˆ ) as follows: Let p observer (see [19], [16]) producing: β(τ be a sufficiently large integer, consider: d ˆ E1 = Eˆ2 + λp+1 (E − Eˆ1 ) dτ d ˆ ˆ ))uav + ξ1 + λp (E − Eˆ1 ) E2 = α(E1 , E2 , I(τ dτ d ξ1 = ξ2 + λp−1 (E − Eˆ1 ) dτ .. .

d ξp−1 dτ d ξp dτ

= λ0 (E − Eˆ1 )

The total stored energy estimation error E˜ = E − E1 , is governed by (21)

The following combination of active disturbance rejection controller and a certainty equivalence controller uav

=

v

=

ˆ ) v − β(τ ˆ )) α(E1 , E2 , I(τ E¨∗ (τ ) − k2 (Eˆ2 − E˙ ∗ (τ )) − k1 (E − E ∗ (τ )) Z t −k0 (E − E ∗ (σ))dσ (22) 0

ˆ ) + k2 E˜˙ e(3) + k2 e¨ + k1 e˙ + k0 e = β(τ ) − β(τ

(23)

Since the right hand side is uniformly absolutely bounded by a small as desired neighborhood of zero, the tracking error e asymptotically exponentially converges to be an uniformly, absolutely, ultimately bounded signal confined to a small as desired vicinity of the origin of the tracking error phase space. D. A Sigma-Delta Modulation Implementation The average controller design generates, under appropriate trajectory planning for the desired total average stored energy, a bounded control input, uav ∈ (0, 1). A switched implementation of the average designed controller is achieved by means of the insertion of a Sigma-Delta modulator, whose output, u, actively takes values in the binary set: {0, 1}. The Sigma-Delta modulator is specified as follows: 1 (1 + sign(e)) (24) 2 Clearly, a sliding motion exists on, e = 0, provided uav (t) ∈ (0, 1). Indeed, as it is easily verified: e˙ = uav (t) − u, u =

ee˙ =

e [uav − 0.5(1 + sign(e))] 1 = − |e| [1 + sign(e)(1 − 2uav (t))] < 0 (25) 2 Ideally, e˙ = 0, implies that the equivalent control, ueq , satisfies ueq = uav . Thus the Sigma Delta modulator provides a switched control input whose average (equivalent) value coincides with the average control input to the modulator. The zero dynamics (ideal sliding dynamics) of the overall system, including the inserted Sigma-Delta modulator, ideally coincides with the desired behavior of the average controlled system (see [20] for details and lab implementations). IV. S IMULATION R ESULTS

= ξp + λ1 (E − Eˆ1 )

E˜(p+2) + λp+1 E˜(p+1) + · · · + λ0 E˜ = β (p) (τ )

produces an average closed loop, tracking error system, e = E − E ∗ (t) governed by

Simulations were carried out for the actual (non-normalized) state and energy variables, in the natural time scale, using a Sigma-Delta modulator implementation of the average GPI observer-based controller design. It was desired to track a given profile of total stored energy smoothly increasing from an initial constant value, Einit = 0.1163[V2 − Farad], towards a final desired constant level, Ef inal = 0.5616[V2 −Farad], in a finite interval of time [7, 22][ms], The smooth increase is to be achieved via an interpolating B´ezier polynomial, θ(t, t0 , T ) such that θ(t0 , t0 , T ) = 0, θ(T, t0 , T ) = 1. In actual (non-normalized) terms with t0 = 7 [ms] and tf inal = 22 [ms], corresponding, respectively, to the normalized instants τ0 = 15 and T = 45. The initial and final normalized energy values, above, correspond, respectively, to initial and final actual output voltages of Vinit = 40.0 [Volts] and Vf inal = 60.0 [Volts]. Simulations were performed on a perturbed Boost converter with the following parameter values L = 20mH, C = 12µF, R = 24.49 Ω, E = 20 V.

6

8 [V Farad]

4 2 0

0.6 0.4

0.01

0.2 0 0

0.02

0.01

time [s]

80

[V]

60

0 0

2

u(t), uav (t) 1

vC (t) 0.01

Vinit

0

0.02

1 0

0.01

time [s] [normalized units]

60

[A]

0.2

0.2 0

ˆ I(t), I(t) 0.01

0.02

time [s]

0.4

0

0.02

time [s]

Vfinal

40 20

R EFERENCES

E(t), E ∗ (t)

2

[A]

6

iL (t)

0.02

40

ˆ β(t), β(t)

20 0 20 0

0.01

time [s]

0.02

time [s]

Fig. 2. Performance of GPI observer-based Total Disturbance Rejection plus Flatness based controller

The unknown (normalized) disturbance load current, I(τ ), was specified by the following time function: 2

I(t) = 0.2(1 + sin(390πt) cos(260πt)e− cos

(195πt)

) [A]

Figure 2 depicts the performance of the GPI observer in the estimation of the load current, I(t), as well as that of the proposed flatness based active disturbance rejection controller in the time responses of the inductor current, iL , and of the output voltage, vC , during the smooth energy transition. The output voltage, vC (t), shows the typical non-minimum phase response; which initially decreases when an increase is demanded.2 V. C ONCLUSIONS In this article, we have explored the effectiveness of a combined control design approach, using total active disturbance rejection and differential flatness, for the regulation of the total stored energy in a boost converter, subject to significant timevarying load demands. As an added bonus, the output voltage is kept nearly constant around the nominal equilibrium value corresponding to the desired value of the total stored energy. The approach was validated through simulations involving significantly time varying load current demands on the part of an unknown resistive load. The approach may be extended to other converter circuit topologies. 2 A high gain observer, such as the GPI observer, tends to produce estimates with large “peakings”. We use a“clutching” operation on the observer variables. This entitles a suitable smoothing factor, SF (t, δ), during a small time interval of length δ. We use, SF (t, δ) = sinq (πt/(2δ)), t ≤ δ and SF (t, δ) = 1, t > δ, with q being an even integer, say, 8.

[1] S.R.H. Amrei, D. Guo-Xu and Y.Q. Lang, ”A New Approach to Hi-Power DC-DC Converters Under Large Load and Input Voltage Variations ”, International Conference on Power Electronics and Drives Systems, PEDS, 2005, pp. 152-157. [2] C.Y. Chan, ”Parallel damped passivity-based control of quasi-resonant converters”, IEE Proceedings - Electric Power Applications, vol. 152, No. 5, Septembre 2005, pp. 1093-1099. [3] S. Diop and M. Fliess, “Nonlinear observability, identifiability and persistent trajectories,” in Proceedings of the 36th IEEE Conference on Decision and Control, Brighton, England, 1991, pp. 714 – 719. [4] M. Fliess and C. Join, “Intelligent PID controllers,” in Control and Automation, 2008 16th Mediterranean Conference on, 25-27 2008, pp. 326 –331. [5] M. Fliess, C. Join, and H. Sira-Ramirez, “Non-linear estimation is easy,” International Journal of Modelling, Identification and Control, vol. 4, no. 1, pp. 12–27, 2008. [6] M. Fliess, J. L´evine, P. Martin, and P. Rouchon, “Flatness and defect of non-linear systems: introductory theory and applications,” International Journal of Control, vol. 61, pp. 1327–1361, 1995. [7] M. Fliess and C. Join, “Commande sans mod`ele et commande a` mod`ele restreint,” e-STA, vol. 5, 2008, available at http://hal.inria.fr/inria00288107/en/. [8] Z. Gao, Y. Huang, and J. Han, “An alternative paradigm for control system design”, 40th IEEE Conference on Decision and Control, vol. 5, pp. 4578-4585, 2001. [9] Z. Gao, “Active disturbance rejection control: A paradigm shift in feedback control system design”, American Control Conference, pp. 2399-2405, 2006. [10] J. Han, “From PID to active disturbance rejection control,” IEEE Transactions on Industrial Electronics, vol. 56, no. 3, pp. 900–906, March 2009. [11] S. Hiti and D. Boroyevich, ”Control of Boost Converter with Adjustable Output Voltage and Unknown Resistive Load”, Power Electronics Specialists Conference, PESC., 25th Annual IEEE, 1994, pp. 294-300. [12] G. J. Jeong, I. H. Kim and Y.I. Son, ”Application of Simple Adaptive Control to a DC/DC Boost Converter with Load Variation”, ICROS-SICE International Joint Conference, Japan, 2009, pp. 1747-1751. [13] J. Liu, W. Ming and F. Gao, ”A new control strategy for improving performance of boost DC/DC converter based on input-output feedback linearization ”, 8th World Congress on Intelligent Control and Automation (WCICA), 2010, pp. 2439-2444. [14] C. D. Johnson, “Accommodation of external disturbances in linear regulator and servomechanism problems,” IEEE Transactions on Automatic Control, vol. AC-16, no. 6, December 1971. [15] C. D. Johnson, “Real-time disturbance-observers; origin and evolution of the idea. part 1: The early years,” in 40th Southeastern Symposium on System Theory, New Orleans, LA, USA, March 2008. [16] A., Luviano-Ju´arez, J., Cort´es-Romero, and H. Sira-Ram´ırez, “Synchronization of chaotic oscilators by means of proportional integral observers,” International Journal of Bifurcation and Chaos, vol. 20, no. 5, pp. 1509–1517, May 2010. [17] H. Sira-Ram´ırez and S. Agrawal, Differentially flat systems. Marcel Dekker Inc., 2004. [18] H. Sira-Ram´ırez, C. N´un˜ ez and N. Visairo, “Robust sigma-delta generalised proportional integral observer based control of a ’buck’ converter with uncertain loads”, International Journal of Control, Vol. 83, No. 8, pp. 1631-1640, 2009. [19] H. Sira-Ram´ırez, M. Ram´ırez-Neria, and A. Rodr´ıguez-Angeles, “On the Linear Control of Nonlinear Mechanical Systems”, 49th IEEE Conference on Decision and Control, Atlanta, Georgia, USA, December 15-17, 2010. [20] H. Sira-Ram´ırez and R. Silva-Ortigoza, Control Design Techniques in Power Electronics Devices, Springer-Verlag, Power Systems Series, London, 2006. [21] H. Sira-Ram´ırez, M. Velasco-Villa and A. Rodr´ıguez-Angeles, “Trajectory tracking control of an input delayed monocycle”, 2009 American Control Conference, Baltimore, MD., June 2009. [22] B. Sun and Z. Gao, “A DSP-based active disturbance rejection control design for a 1-kW H-bridge DC-DC power converter”, IEEE Transactions on Industrial Electronics, vol. 52, no. 5, pp. 1271-1277, 2005. [23] D. Sun, “Comments on Active Disturbance Rejection Control” IEEE Transactions on Industrial Electronics, Vol. 54, No. 6, pp. 3428-3429, December 2007.

On the Robust Control of the Boost Converter H. Sira-Ram´ırez †, M. A. Oliver-Salazar‡, J.A. V´azquez-Santacruz†, M. Velasco-Villa† †Centro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico Nacional. Departamento de Ingenier´ıa El´ectrica, Secci´on de Mecatr´onica. Apartado postal 14740, M´exico DF C.P. 07360 ‡Centro Nacional de Investigaci´on y Desarrollo Tecnol´ogico Coordinaci´on de Mecatr´onica Cuernavaca, Morelos, M´exico

Abstract—In this article, we propose a linear Total Active Disturbance Rejection approach for the flatness based control of an uncertain “Boost” DC-to-DC power converter subject to unknown load current demands of arbitrary time-varying nature. The approach makes use of two Generalized Proportional Integral (GPI) observers; one for the precise determination of the load current, and a second one for the on-line determination of the additive nonlinearities affecting the stored energy second order dynamics. After disturbance suppression, a certainty equivalence controller completes the controller design by disturbancedependent control input gain cancelation. Simulations demonstrate the effectiveness of the control approach in the tracking of a desired stored energy profile.

I. I NTRODUCTION Controllers based on disturbance estimation, followed by its direct cancelation from the control actions, have been developed along different, but closely related, directions. In this respect, the outstanding work of professor C.D. Johnson known as Disturbance Accommodation Control (DAC), dates from the nineteen seventies (see [14]). The evolution of the DAC approach is surveyed in [15]. Related developments, known as Active Disturbance Rejection (ADR), are addressed in the work of the late professor Jingqing Han [10]. This includes nonlinear observers, with side developments related to: efficient time derivatives computation, practical relative degree calculation and nonlinear PID control extensions. Professor Z. Gao, and his colleagues (see [8], [9], [22], [23]) have made important contributions in ADR and have proposed a new control paradigm, closely related to the vision of Prof. Han. Prof. M. Fliess and C. Join (see [4] and [7]), proposed Intelligent PID Control (IPIDC) related to Prof. Han’s handling of state dependent perturbation inputs. The method is based on Differential Algebra and it implies to resort to low order, phenomenological plant models. The effect of nonlinear uncertain additive terms, locally modeled as timepolynomials that need to be updated, is suitably annihilated. Time-varying state dependent control gains are deemed as piece-wise constant gains which can be efficiently on line identified (see [5] for a sound theoretical basis). Recently, an entirely robust linear approach has been proposed for the control of uncertain non-linear differentially flat systems, using an input-output approach (see [18]) which may be considered as the analog counterpart of the algebraic methods stated in [5]. Linear GPI observers are used to locally estimate, in

a self-updating manner, the joint effects of exogenous and endogenous additive disturbances, while the control input gain is assumed to be known and, at most, output dependent ([19]). The approach, however, has significant implications in nonlinear chaotic systems estimation as demonstrated in [16]. The GPI observer based approach has also been extended to the control of nonlinear multi-variable systems exhibiting known time delays in the control inputs (see [21]). In several boost converter applications, load current variations need special attention. In [12], an adaptive controller is designed to provide robustness against perturbations of the output voltage. In [11] a voltage compensator is developed based on extended linearization and gain scheduling with a nonlinear current mode control. A control strategy is presented in [1] based on the combination of sliding mode control and the current mode control for line and load regulation with low current ripple. In [2] a parallel damping (instead of series damping) passivity-based control stategy is presented to achieve robustness against load variations in the buck and boost converters. The study presented in [13] is an input-output feedback linearization based scheme, taking the inductor current as the controlled output. In this article, we propose a GPI observer-based ADR approach for controlling a perturbed “Boost” DC-to-DC power converter in a total stored energy trajectory tracking task with a time-varying, unknown, load demand. We design a robust GPI observer-based controller, with certainty equivalence input gain cancelation, using the average converter model. The average design is translated into a switched controlled design using the Sigma-Delta modulation implementation alternative. Under the assumption of full state measurement, a GPI observer estimates, in a rather precise fashion, the unknown load current. This estimation is carried out on the basis of the output voltage dynamics. The nonlinear additive terms affecting the total stored energy second order dynamics are also linearly estimated using a GPI observer. The load current dependent control input gain is canceled within a certainty equivalent PID controller scheme. Section II deals with a general formulation of the ADR control problem in nonlinear differentially flat systems. In section III, we present the GPI observer based ADR for the total stored energy trajectory tracking task in a Boost converter subject to unknown load current demands. Section IV presents simulation results assessing the effectiveness of the proposed

2

control scheme. The conclusions and suggestions for further research are collected at the end of the article. II. ACTIVE D ISTURBANCE R EJECTION IN N ONLINEAR D IFFERENTIALLY F LAT S YSTEMS Consider the single-input, single-output, smooth nonlinear n-dimensional input-output perturbed system, y (n)

=

˙ ψ(t, y, y, ˙ ..., y (n−1) , ζ(t), ζ(t)) +φ(t, y, y, ˙ · · · , y (n−1) , ζ(t))u

(1)

The unperturbed system is, therefore, differentially flat, with flat output, y. We say that the system is perturbed differentially flat, whenever ζ(t) is not identically zero. In this last case, the state components in the vector, x, and the control input, u, are differentially parameterizable in terms of, both, the output, y, and the exogenous perturbation input ζ(t). Conversely, the phase variables (y, y, ˙ ..., y (n−1) ) may be written in terms of the n-dimensional state of the system, denoted by x, the perturbation input, ζ, and a finite number of its time derivatives. The above input-output description of the system typically corresponds to non-matched disturbances in the state space formulation1 . (n−1) ˙ The term: ψ(t, y(t), y(t),...,y ˙ (t), ζ(t), ζ(t)), is assumed to be uniformly absolutely bounded for any suitable control input, u, achieving the control objectives. The nonlinear gain function, φ(t, y(t), y(t), ˙ · · · , y (n−1) (t), ζ(t)), is known and also assumed to be bounded and uniformly bounded away from zero, i.e., there exists a strictly positive constants, µ, and, β, such that: supt |φ(t, y(t), y(t), ˙ · · · , y (n−1) (t), ζ(t))| ≤ β and µ ≤ inf t |φ(t, y(t), y(t), ˙ · · · , y (n−1) (t), ζ(t))|, for all smooth, bounded solutions, y(t), of (1) obtained with a suitable control input u achieving the control objectives. We formulate the problem as follows: Given a desired flat output reference trajectory, y ∗ (t), devise a linear output feedback controller for system (1) so that, regardless of the effect of the additive perturbation signal, ψ(·), the flat output, y, tracks the desired reference signal, y ∗ (t), even if in an approximate fashion. This approximate character specifically means that the tracking error, e(t) = y−y ∗ (t), and its first n-time derivatives, converge towards a small as desired neighborhood of the origin in the reference trajectory tracking error phase space in a globally asymptotically exponentially dominated manner. We denote by, ξ(t), the following expression, ξ(t) = ψ(t, y(t), y(t), ˙ ..., y (n−1) (t), ζ(t)) This quantity, deemed as a perturbation input, needs to be online estimated for proper subsequent cancelation at the controller stage. The signal, ξ(t), viewed from the observer design perspective, represents an exogenous time-varying quantity which is easily shown to be observable in the sense of 1 Specifically the state representation is of the form: x ˙ = f (x) + g(x)u + h(x) 6= γ(x)ζ, y = h(x) with Lg Ljf h(x) = 0, j = 1, 2, ..., n−2, Lg Ln−1 f 0 on a sufficiently large open region of Rn containing an operating point, x0 . Further, Lγ Lkf h(x) = 0, k = 0, 1, ..., n − 3, Lγ Lif h(x) 6= 0, i = n − 2, n − 1. This is, precisely, the case of the Boost converter here treated.

Diop and Fliess [3]. Regarding this state dependent signal as an, unstructured, time-varying signal, is at the heart of the observer-based ADR approach. Our linear observer design strategy consists in approximately estimating this time-varying quantity, ξ(t), using an, instantaneous, internal time polynomial model, realized in the form of a chain of integrators of length, p − 1, at the observer stage for a fixed, sufficiently large integer p. The observer will also approximately estimate, in a simultaneous fashion, the phase variables associated to the system flat output y. When forcing the dominantly linear, perturbed, output estimation error dynamics to exhibit an asymptotically convergent behavior, the internal model for ξ(t) is automatically and continuously self-updated. We assume that ξ(t) and a finite number of its time derivatives, ξ (k) (t), are uniformly absolutely bounded, for k = 1, 2, ..., p, for some sufficiently large integer p. We have the following general result: Theorem 1: The GPI observer-based dynamical feedback controller: h u=

[y ∗ (t)](n) −

Pn−1 ¡ j=0

i ¢ ˆ κj [yj − (y ∗ (t))(j) ] − ξ(t)

φ(t, y, y, ˙ · · · , y (n−1) , ζ(t))

ˆ = z1 ξ(t) y˙ 1 = y2 + λp+n−1 (y − y1 ) y˙ 2 = y3 + λp+n−2 (y − y1 ) .. . y˙ n = φ(t, y)u + z1 + λp (y − y1 ) z˙1 = z2 + λp−1 (y − y1 ) .. . z˙p−1 = zp + λ1 (y − y1 ) z˙p = λ0 (y − y1 )

(2)

asymptotically exponentially drives the tracking error phase (k) variables, ey = y (k) − [y ∗ (t)](k) , k = 0, 1, .., n − 1 to an arbitrary small neighborhood of the origin, of the tracking error phase space, which can be made as small as desired from the appropriate choice of the controller gain parameters, {κ0 , ..., κn−1 }. Moreover, the estimation errors: e˜(i) = y (i) − yi+1 , i = 0, ..., n − 1 and the perturbation estimation errors: zj − ξ (j−1) (t), j = 1, ..., p, asymptotically exponentially converge towards a small as desired neighborhood of the origin of the reconstruction error space, with the appropriate choice of the observer gain parameters, {λ0 , ..., λp+n−1 }. Proof The proof is based on the fact that the estimation error e˜ satisfies the perturbed linear differential equation e˜(p+n) + λp+n−1 e˜(p+n−1) + · · · + λ0 e˜ = ξ (p) (t)

(3)

Since ξ (p) (t) is assumed to be uniformly absolutely bounded then there exists coefficients λk such that e˜ converges to a small vicinity of zero, provided the roots of the associated characteristic polynomial in the complex variable s: sp+n + λp+n−1 sp+n−1 + · · · + λ1 s + λ0

(4)

3

are all located deep into the left half of the complex plane. The further away from the imaginary axis, of the complex plane, are these roots located, the smaller the neighborhood of the origin, in the estimation error phase space, where the estimation error e˜ will remain uniformly ultimately bounded. Let l(˜ e, e˜˙ , ...˜ e(n−1) ) be a linear function of the estimation error, e˜, and its time derivatives, given by l(˜ e, e˜˙ , ...˜ e(n−1) ) =

n−1 X

kj e˜(j)

(5)

L

iL

E

Fig. 1.

u=1

vC

u=0

I(t)

C

R

Perturbed Boost Converter Circuit

j=0

The tracking error, ey = y − y ∗ (t), evolves according to the following linear perturbed dynamics (n−1) e(n) + · · · + κ0 ey y + κn−1 ey ˆ − l(t, e˜, e˜˙ , ...˜ = ξ(t) − ξ(t) e(n−1) )

Choosing the controller coefficients {κ0 , · · · , κn−1 }, so that the associated closed loop characteristic polynomial sn + κn−1 sn−1 + · · · + κ0

(6)

exhibits its roots sufficiently far from the imaginary axis in the left half portion of the complex plane, the tracking error, and its various time derivatives, are guaranteed to dominantly asymptotically exponentially converge towards a vicinity of the origin of the tracking error phase space. We also have that the disturbance estimation error satisfies: ξ(t) − ξˆ = ξ(t) − z1 = r(t) = e˜(n) + λp e˜ Since e˜(n) and e˜ are forced to converge to a small as desired vicinity of the origin, the local residual term, r(t), of the polynomial approximation is uniformly absolutely bounded by a small neighborhood of zero. This fact clearly depicts the selfupdating character of the local time polynomial model of the lumped disturbance ξ(t).

where, x1 , is the normalized inductor current. p The variable, y = x2 , is the normalized output voltage, Q = R C/L, is the fixed normalized resistive output load, assumed to be perfectly known, and I(t) denotes the normalized time-varying load current. When, with an abuse of notation, the normalized equations (8) are written, substituting the switched input u by the continuous input, uav ∈ (0, 1), we refer to the controlled system as the average system model. The corresponding state variables x1 , x2 and the output y are deemed as the average state variables. We specifically state the following set of assumptions on the average system, 1) I(τ ) is strictly positive and uniformly bounded. Its first ˙ ), is continuous and uniformly order time derivative, I(τ absolutely bounded. 2) The normalized resistive load Q is constant and perfectly known. All other system parameters allowing the above normalization are perfectly known 3) The average input current, x1 , and the average output voltage y = x2 are measurable. We deal with the total normalized stored energy of the system, denoted by E, defined as

III. ROBUST C ONTROL OF THE T OTAL S TORED E NERGY IN A B OOST C ONVERTER

1 2 (x + x22 ) (9) 2 1 The total power, i.e., the first order time derivative of, E, is readily found to be given by

Consider the Boost converter system, shown in Figure 1, subject to an unknown, time-varying, external load current demand, I(t). The system equations are readily obtained as

x2 E˙ = x1 − 2 − x2 I(τ ) Q

diL dt dvC C dt η L

=

−uvC + E

=

uiL −

=

vC

vC − I(t) R

(7)

iL is the inductor current, vC is the output voltage, u ∈ {0, 1} is switched control input. η is the output voltage. All system parameters L, C, R and E are assumed to be perfectly known. We perform the following normalizing p state, and time, coordinate transformation: x = (i/E) L/C, x2 = v/E, 1 √ τ = t/ LC. The normalized equations, governing the converter, are given by (“ ˙ ” stands for d/dτ ): x˙ 1

=

x˙ 2

=

y

=

−ux2 + 1 x2 − I(τ ) ux1 − Q x2

(8)

E=

(10)

where the first term represents the input power (the product of the normalized input current and the unit normalized voltage source) the second term is the power dissipated in the resistive load and, the last term, represents the input power to the unknown non-regenerative load. The total stored energy variable, E, is relative degree 2 and, via some straightforward algebraic manipulation, is seen to satisfy the following average perturbed relation: ¸ · 2x1 x2 ¨ + x1 I(τ ) uav E = − x2 + Q 2x2 3x2 I(τ ) ˙ ) (11) +1 + 22 + + I 2 (t) − x2 I(τ Q Q The unperturbed Boost converter is differentially flat (see Fliess et al. [6] and, also, Sira-Ram´ırez and Agrawal [17]) with the total stored energy, E, as the flat output. The perturbed version of the system is differentially parameterizable in terms

4

of the flat output, the perturbation input and a finite number of its time derivatives. Clearly, the average total stored energy E satisfies a second order system of perturbed differential equations of the form: E˙1 E˙2

= =

E2 ˙ )) (12) α(E1 , E2 , I(τ ))uav + β(E1 , E2 , I(τ ), I(τ

with E1 = E and E2 = E˙ and, in terms of the states x1 , x2 , · ¸ 2x1 x2 ˙ α(E, E, I(τ )) = − x2 + + x1 I(τ ) Q 2 ˙ I(τ ), I(τ ˙ )) = 1 + 2x2 + 3x2 I(τ ) + I 2 (τ ) β(E, E, 2 Q Q ˙ −x2 I(τ ) The lack of knowledge of I(τ ), the need to compute E˙ and the difficulty in on-line synthesizing the nonlinearities α(·) and β(·) prompts us to formulate the problem as follows: A. Problem Formulation Given the average Boost converter circuit, (8), where all the previous assumptions are valid. It is desired to smoothly regulate the average total stored energy, E, from a given initial constant value: Einit , towards a final desired value, Ef inal , in a pre-specified amount of normalized time, T −τ0 , irrespectively of the, time-varying, magnitude of the load demand I(τ ) and based only on synthesis of the total stored energy E. B. A GPI Observer for the Load Current Consider the uncertain average output voltage equation with, y, representing the measured output voltage x2 . y y˙ = uav x1 − − I(τ ) (13) Q Definition 2: A signal, z(τ ), is said to be ultimately arbitrarily close to a signal, ζ(τ ), denoted by z(τ ) ≈ ζ(τ ), whenever given a small as desired constant, ² > 0, for which there exists a finite time, t² > 0, such that, the error signal, e(τ ) = z(τ ) − ζ(τ ), satisfies: supτ ≥τ² |e(τ )| ≤ ². In other words, the error e(τ ) is said to be uniformly absolutely ultimately bounded by an open spherical neighborhood of zero, of radius ². We have the following result: Lemma 3: Let m be a sufficiently large integer. The GPI observer, yˆ yˆ˙ = uav x1 − + z1 + γm (y − yˆ) Q z˙1 = z2 + γm−1 (y − yˆ) z˙2 = z3 + γm−2 (y − yˆ) .. . z˙m

= γ0 (y − yˆ)

(14)

produces a redundant estimate, yˆ, of the measured output variable, y, and an estimate, −z1 , of the unknown, timevarying, load current I(τ ) which are ultimately arbitrarily close to the their actual values, y(τ ), and, I(τ ), provided

the set of coefficients: {γ0 , . . . , γm }, of the characteristic polynomial pI (s), given by: 1 pI (s) = sm+1 + (γm + )sm + γm−1 sm−1 + · · · + γ1 s + γ0 Q (15) are chosen such that the roots of, pI (s), are located sufficiently far into the left half of the complex plane. Proof (m) Let ey = e1 = y − yˆ, e˙ y = e2 = y˙ − yˆ˙ ,..., ey = em+1 = (m) (m) y − yˆ denote the phase variables of the reconstruction error dynamics. From (13) and (14), the estimation, or reconstruction, error vector dynamics is then given by 1 e˙ 1 = −z1 − I(τ ) − (γm + )e1 Q z˙1 = z2 + γm−1 e1 .. . z˙m = γ0 e1 (16) Upon elimination, through straightforward differentiation, of the z variables in (16), the output voltage reconstruction error ey = y − yˆ is readily seen to satisfy the following linear, injected, perturbed, scalar differential equation: 1 e(m+1) + (γm + )e(m) + γm−1 e(m−1) + ··· y y Q y (17) · · · + γ1 e˙ y + γ0 ey = −I (m) (τ ) Since, by assumption I (m) (τ ) is L∞ then there exists a strictly positive constant K such that supt |I (m) (τ )| ≤ K. Define the (m) vector: χ = (ey , e˙ y , · · · , ey )T . The disturbed system has the (m) form: χ˙ = Aχ + bI (τ ), with A being a stable matrix in companion form, and b a vector with zero elements, except for its last component which has value equals to −1. The matrix, Q = A + AT , is symmetric and negative definite, with real negative eigenvalues, denoted as λ(Q). Let λmax (Q) be the eigenvalue with the minimum absolute value. The Lyapunov function candidate, V (x) = 21 χT χ = 12 kχk2 , satisfies: 1 T χ (A + AT )χ − χT I (m) (τ ) 2 1 ≤ − |λmax (Q)|kχk2 + Kkχk 2 The spherical region: V˙ (χ) =

S = {χ ∈ Rm+1 | kχk2 ≤ 4K 2 /|λmax (Q)|2 },

(18)

defines the convergence region of all the reconstruction error trajectories in the reconstruction error phase space, (ey , e˙y , · · · , ey (m) ). As the distance between the eigenvalues of the matrix A (i.e, the roots of p(s)) and the imaginary axis is increased, the radius of the sphere S is reduced, reducing the region of convergence of the injected reconstruction error trajectories. Consider now the equation satisfied by ey in terms of z1 . We have 1 e˙ 1 = −z1 − I(τ ) − (γm + )e1 (19) Q Since e1 = ey and e˙ 1 = e2 are ultimately absolutely bounded by a small as desired ball around the origin, then, from the

5

1 above relation, z1 = −I(τ ) − (γm + Q )e1 − e2 . Hence, the difference between z1 and −I(τ ) is also uniformly absolutely ultimately bounded by a small as desired sphere located at the origin. z1 is then ultimately arbitrarily close to −I(τ ). By the same token, from the differential equation for z2 , namely: ˙ )−γm−1 e1 . Since z˙1 = z2 +γm−1 e1 , it follows that z2 ≈ −I(τ e1 is uniformly ultimately bounded to a small as desired region of the origin of the estimation error phase space, it follows that ˙ ). In general zj (τ ) ≈ z2 is ultimately arbitrarily close to −I(τ (j−1) −I (τ ), j = 1, 2, ... in the above sense. Arbitrarily close estimation of the unknown load current −I(τ ) allows the use of its arbitrarily close estimate z1 in a certainty equivalence controller design for the total stored energy E.

C. A Linear GPI Observer Based Active Disturbance Rejection Controller for the Total Stored Energy Consider the average total stored energy dynamics, E˙1 E˙2

= =

E2 (20) ˙ )) α(E1 , E2 , I(τ ))uav + β(E1 , E2 , I(τ ), I(τ

Since E, and its time derivative, are known, thanks to state availability and the GPI observer based gathered knowledge of the perturbation input current demand, I(τ ); the control ˙ I(τ )), can be efficiently canceled from input gain, α(E, E, ˆ )). The additive the certainty equivalence gain: α(E1 , E2 , I(τ ˙ )), may be considered perturbation term, β(E1 , E2 , I(τ ), I(τ to be unknown, and being an exogenous signal to the GPI ˙ )), can observer, its time realization, β(E1 (τ ), E2 (τ ), I(τ ), I(τ be linearly estimated, in a lumped manner, using a GPI ˆ ) as follows: Let p observer (see [19], [16]) producing: β(τ be a sufficiently large integer, consider: d ˆ E1 = Eˆ2 + λp+1 (E − Eˆ1 ) dτ d ˆ ˆ ))uav + ξ1 + λp (E − Eˆ1 ) E2 = α(E1 , E2 , I(τ dτ d ξ1 = ξ2 + λp−1 (E − Eˆ1 ) dτ .. .

d ξp−1 dτ d ξp dτ

= λ0 (E − Eˆ1 )

The total stored energy estimation error E˜ = E − E1 , is governed by (21)

The following combination of active disturbance rejection controller and a certainty equivalence controller uav

=

v

=

ˆ ) v − β(τ ˆ )) α(E1 , E2 , I(τ E¨∗ (τ ) − k2 (Eˆ2 − E˙ ∗ (τ )) − k1 (E − E ∗ (τ )) Z t −k0 (E − E ∗ (σ))dσ (22) 0

ˆ ) + k2 E˜˙ e(3) + k2 e¨ + k1 e˙ + k0 e = β(τ ) − β(τ

(23)

Since the right hand side is uniformly absolutely bounded by a small as desired neighborhood of zero, the tracking error e asymptotically exponentially converges to be an uniformly, absolutely, ultimately bounded signal confined to a small as desired vicinity of the origin of the tracking error phase space. D. A Sigma-Delta Modulation Implementation The average controller design generates, under appropriate trajectory planning for the desired total average stored energy, a bounded control input, uav ∈ (0, 1). A switched implementation of the average designed controller is achieved by means of the insertion of a Sigma-Delta modulator, whose output, u, actively takes values in the binary set: {0, 1}. The Sigma-Delta modulator is specified as follows: 1 (1 + sign(e)) (24) 2 Clearly, a sliding motion exists on, e = 0, provided uav (t) ∈ (0, 1). Indeed, as it is easily verified: e˙ = uav (t) − u, u =

ee˙ =

e [uav − 0.5(1 + sign(e))] 1 = − |e| [1 + sign(e)(1 − 2uav (t))] < 0 (25) 2 Ideally, e˙ = 0, implies that the equivalent control, ueq , satisfies ueq = uav . Thus the Sigma Delta modulator provides a switched control input whose average (equivalent) value coincides with the average control input to the modulator. The zero dynamics (ideal sliding dynamics) of the overall system, including the inserted Sigma-Delta modulator, ideally coincides with the desired behavior of the average controlled system (see [20] for details and lab implementations). IV. S IMULATION R ESULTS

= ξp + λ1 (E − Eˆ1 )

E˜(p+2) + λp+1 E˜(p+1) + · · · + λ0 E˜ = β (p) (τ )

produces an average closed loop, tracking error system, e = E − E ∗ (t) governed by

Simulations were carried out for the actual (non-normalized) state and energy variables, in the natural time scale, using a Sigma-Delta modulator implementation of the average GPI observer-based controller design. It was desired to track a given profile of total stored energy smoothly increasing from an initial constant value, Einit = 0.1163[V2 − Farad], towards a final desired constant level, Ef inal = 0.5616[V2 −Farad], in a finite interval of time [7, 22][ms], The smooth increase is to be achieved via an interpolating B´ezier polynomial, θ(t, t0 , T ) such that θ(t0 , t0 , T ) = 0, θ(T, t0 , T ) = 1. In actual (non-normalized) terms with t0 = 7 [ms] and tf inal = 22 [ms], corresponding, respectively, to the normalized instants τ0 = 15 and T = 45. The initial and final normalized energy values, above, correspond, respectively, to initial and final actual output voltages of Vinit = 40.0 [Volts] and Vf inal = 60.0 [Volts]. Simulations were performed on a perturbed Boost converter with the following parameter values L = 20mH, C = 12µF, R = 24.49 Ω, E = 20 V.

6

8 [V Farad]

4 2 0

0.6 0.4

0.01

0.2 0 0

0.02

0.01

time [s]

80

[V]

60

0 0

2

u(t), uav (t) 1

vC (t) 0.01

Vinit

0

0.02

1 0

0.01

time [s] [normalized units]

60

[A]

0.2

0.2 0

ˆ I(t), I(t) 0.01

0.02

time [s]

0.4

0

0.02

time [s]

Vfinal

40 20

R EFERENCES

E(t), E ∗ (t)

2

[A]

6

iL (t)

0.02

40

ˆ β(t), β(t)

20 0 20 0

0.01

time [s]

0.02

time [s]

Fig. 2. Performance of GPI observer-based Total Disturbance Rejection plus Flatness based controller

The unknown (normalized) disturbance load current, I(τ ), was specified by the following time function: 2

I(t) = 0.2(1 + sin(390πt) cos(260πt)e− cos

(195πt)

) [A]

Figure 2 depicts the performance of the GPI observer in the estimation of the load current, I(t), as well as that of the proposed flatness based active disturbance rejection controller in the time responses of the inductor current, iL , and of the output voltage, vC , during the smooth energy transition. The output voltage, vC (t), shows the typical non-minimum phase response; which initially decreases when an increase is demanded.2 V. C ONCLUSIONS In this article, we have explored the effectiveness of a combined control design approach, using total active disturbance rejection and differential flatness, for the regulation of the total stored energy in a boost converter, subject to significant timevarying load demands. As an added bonus, the output voltage is kept nearly constant around the nominal equilibrium value corresponding to the desired value of the total stored energy. The approach was validated through simulations involving significantly time varying load current demands on the part of an unknown resistive load. The approach may be extended to other converter circuit topologies. 2 A high gain observer, such as the GPI observer, tends to produce estimates with large “peakings”. We use a“clutching” operation on the observer variables. This entitles a suitable smoothing factor, SF (t, δ), during a small time interval of length δ. We use, SF (t, δ) = sinq (πt/(2δ)), t ≤ δ and SF (t, δ) = 1, t > δ, with q being an even integer, say, 8.

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